I started a new painting at Montgomery Hill park in East San Jose yesterday, next to Evergreen Valley Community College; the park is named after aviation pioneer John Montgomery who died when his glider, The Evergreen, crashed here in 1911. Sylvia suggests that I might add a painting of a UFO-type Flying Wing to a painting of this hill, like the one I wrote about in my novel, The Secret of Life. By the way the image above is an High Dynamic Range image fused from three exposures.

Word form your sponsor: “Jeez, not a single person has bought a notecard of my paintings at, ahem, rudy.imagekind.com, and they’re only $3.29 each (plus shipping). Whaddaya whaddaya.”

As regular readers of this blog know, I’m currently interested in a new cosmological model called the Cyclic Universe and have posted on it before. The theory has been popularized by the scientists Paul Steinhardt and Neil Turok in their very readable book, The Endless Universe. The basic idea is that our cosmos consists of two parallel sheets of space which are called branes.

Every so often (one guess is every trillion years), the branes slam together, which fills them with energy that can be in some sense gotten for free from the boundless riches of the gravitational field. This space-filling Big Splat replaces the point-like Big Bang. While the branes are apart, they expand, with the galaxies moving apart over the years. And eventually they embrace once again. It could be that there are an endless number of cycles in the past and in the future, and that the spaces involved are infinite.

The theory of the Cyclic Universe is controversial and some well-entrenched phycisists such Stanford’s Andrei Linde (whom I interviewed for Wired , years ago) and his wife Renata Kallosh are fighting against the theory with what I would call excessive or even hysterical or persecutional zeal. You can (vaguely) follow the battle on the boffin-zone arXiv.org site (see this Wikipedia entry on arXiv to learn what arXiv is all about).

In my amateur outsider’s opinion, Steinhardt’s camp is right. LInde’s inflationary theory is falling apart. One of the nicest things favoring Steinhardt’s side is that the Cyclic Universe theory only requires for there to be one (two-braner) universe, whose parameters are what they are due to certain underlying mathematical reaons. The Lindean inflationary camp claims there are perhaps googolplex universes and the one we’re in just happens to have the particular values we observe. Hopefully some new measurements of the cosmic microwave background radiation can help resolve the dispute experimentally before too long.

In any case, I decided I might write some science fiction about the Cyclic Universe, and I sent Paul Steinhardt a few questions about the theory via email. He answered them and kindly offered to talk on the phone so that I could ask some follow-up questions. The following is a distillation of the email and phone conversations. As it’s been edited by me, any errors should be ascribed to Rudy rather than to Paul.

(R1) It would be cool if a story’s main character finds out that the Big Splat of the recolliding branes is going to be TOMORROW, rather than in a trillion years.

(P1) According to our current models, the soonest the next splat could really happen is in ten billion years. The reason is that it takes nearly ten billion years for the branes to move back together, and as soon as they start approaching each other we can notice subtle changes. So if the branes were on the road to colliding, we would have noticed by now.

(R2) What are the kinds of observational evidence that tell you the tenth-dimensionsal separation between branes is dwindling? As I understand it, our universe itself will continue to seem like it’s expanding, so how might we notice the impending splat?

(P2) You’ll like this: as the branes begin to move towards one another, nature’s fundamental “constants” begin to change— Newton’s gravitational constant, the fine-structure constant that controls the strength of electricity and magnetism, etc. And the rate of change picks up the closer the branes get. It would seem like the laws of physics are changing faster and faster.

In the earliest stages, the first thing we’d notice would be slight variations in the spectral lines between nearby atoms and distant (older) atoms. Later on, the changes would be more dramatic. We’d notice the positions of spectral lines in a single sample of matter changing over the course of a day.

Later on, atoms might get larger, but you’d have other effects mixed in as well. It would become in some sense hard to say what size anything is, as our definitions of size are ultimately based on the fundamental constants.

(R3) Presumably any humans would be destroyed at some point; our molecules would fall apart, and so on. And then, of course, you’d have the Big Flash. Might we have any hope of surviving to the next cycle?

(P3) We might draw on the fact that the collisions between the branes only occur in places where the universe is nearly empty — which is ALMOST everywhere. But where there are black holes, their gravitational field is strong enough that the collisions do not occur near them. It is as if the black hole pins the branes together in those places where they lie, and that means there are no collisions.

If an advanced civilization could figure out how to create/manipulate black holes so that they are surrounded by them but do not fall into them and if they could protect themselves from the intense radiation of the collision (about 10^10 times the temperature of the sun), they could survive into the next cycle. One problem here is that, because of all the stretching of space that occurs from cycle to cycle, surviving black holes are spread out to an enormous degree — so the chances are that, by the time the Splat approaches, there are no black holes within mankind’s observable horizon. So, even if a civilization managed to do this, they would not likely be anywhere we could see them.

(R4) I like that idea. Maybe a science-fictional civilization could manufacture black holes! Of course it would be tricky to surround our planet with them and not have the holes collapse together. Maybe they could be furiously orbiting in some chaotic dance.

On a slightly different topic, I’m intrigued by the notion of finding some way to perceive that space is infinite. But we’re to some extent limited by that big flash that happened 14 billion years ago. If the universe is spatially infinite, might there perhaps be SOME kind of signal that makes it through the haze of the Big Flash, reaching us from distances larger than the light horizon? I’m thinking, for instance, of gravitons that started on their way before the last Splat, or even from several Splats ago.

(P4) Yes and no. There would be gravitational waves produced just before the last bang that we could detect in the present universe. But the signal is very, very weak. And I suppose a civilization that existed just before the end could send that signal. But the bigger problem with the gravitational waves and photons from earlier cycles, is that their wavelengths will have been stretched by the expansion of space. The 14 billion light year diameter sphere that we see around us began as a region less than a meter across. A meter-long gravity wave from those times would now span the visible universe. There would be no way to construct an apparatus to detect it.

(R5) So there’s no way to pass information from one cycle to the next?

(P5) One very speculative idea is that you might store information inside a black hole. Although we think of the two branes as parallel with a tiny separation, in the neighborhood of a black hole, as I mentioned before, the two branes dimple out and touch each other, merging into a single brane. So there’s not going to be an big splat and radiation burst inside a black hole.

Of course the hard part about storing information inside a black hole is getting it back out. There’s an ongoing debate among physicists about whether information that goes into a black hole disappears for good, or whether it might be retrievable if the black hole spontaneously evaporates. Or perhaps if you scatter some object off of the black hole. In the scattering case, it may be that you need something like an encryption key to retrieve the information, that is, you have to in some sense know what the black hole has eaten so far.

(R6) In terms of a smooth motion, it seems like it would be nicer if the two branes could pass through each other, rather than splatting and rebounding. Is that a possibility?

(P6) Sure, I like to think of it that way myself, as then you feel less worried about the branes getting stuck together. Whether we say they bounce or pass through each other is really just a matter of how we set up our coordinate system.

(R7) Might there be life on the other brane?

(P7) We think of the other brane as probably not being something like a mirror of our world, so it would be a very different kind of place. One might not have atoms or particles there in the usual sense.

I put some more of my paintings up on Imagekind where you can buy prints of 34 of them and — tada! — you can now buy them as inexpensive greeting cards!

It’s a major hassle getting the pictures into digital form. I reshot some of the older ones too, by the way, to enhance the resolution. I photograph them on high-end slide film with my old “safari model” (i.e. green) Leica R3 single-lens reflex on a tripod, get the film devloped at Superior Color Lab in the Willow Glen neighborhood of San Jose, then send the slides to this nice place, My Special Photos, in Los Altos up the peninsula that scans them at 4,000 dpi to 16-bit deep color TIF files with effectively 24 megapixels per image, using a high-end Nikon scanner.

And then I PhotoShop the hell out of the images to make them pop. Speaking of PhotoShop, I just got CS3 and am playing a little with the HDR (High Dynamic Range) gimmick where you can fuse several images taken at different exposures so as to get better highlights and shadow areas; the image above is made of three exposures. The one below is a better picture, compositionally, using just one exposure, though it’s not all that sharp and maybe I over-PhotoShopped the color. Both shot at Castle Rock Park.

Every step of the way, in turning my paintings into digital files, I’m dogged by uncertainty and worry; the most maddening thing is that at the end of the road, I’ll often feel that the original photo I shot wasn’t as good as it could have been in terms of lighting. Glare is a problem, as is over or under exposure, even when I shoot bracket shots.

In any case, I’ve come to accept that the final colors are more or less arbitrary, with only a casual relation to the original. ‘s all PhotoShoppable. But bad glare or a blown-out negative is hard to compensate for.

My friend Mimi was kidding me about “learning to suffer for my art.” Uploading the 80 megabyte files to Imagekind is a hassle too, I might add, it takes about an hour per file, and doesn’t always work.

I’m feeling a little frazzled these days, as I’ve had a frikkin’ cold for three weeks, which gets depressing. Also I miss the “narcotic moment of creative bliss” that I get from writing. Now and then a photograph gives me a tiny hit. PhotoShop CS3 has this nice FilterDistort|Lens Correction filter that lets you get rid of barrel and keystone distortion in a shot of something rectangular like the frame above. I love neon martini signs.

It would be simpler to shoot my paintings with a heavy-duty SLR digital camera, of course, as then I could right away see what I’d shot, and I wouldn’t have to deal with two separate photo labs. But (a) I don’t have one, and (b) if I did, the current typical 12 megapixel size would be about half of the 24 megapixel I get via the hig-res scan, although it could be that the crispness would be just as good if I was doing it digitally, also I could be sure I got the framing and exposure right. It could well be that the slide to digital conversion puts in more noise than you’d have if you shot straight to digital and just did a resampling to raise the size.

I do love using that old Leica. Actually you can get an adapter to put Leica lenses onto Canon EOS bodies, although then you have to do manual focusing. The lower end bodies are plastic, which I hate the feel of, but you can get a magnesium body Canon 5D for under $2K, or a Canon 1D Mark III with a 20 megabyte sensor for maybe $5K, but that’s really a lot of money to spend on something I might not use that much.

In practice, I really like shooting with my pocket SONY Cybershot 8 Megapixel T100. If I have a camera in my pocket all the time, I get lots of shots, and if I have to put pounds-heavy camera around my neck, I hardly use it. And, after all, mostly I just use my photos for my blog at some really low pixel count, like 2 or 3 megapixels. Of course if I got into selling prints I could open up a whole new world of effort…

Anyway here’s descriptions of the five latest paintings of mine that you can buy prints of.

31. Mossy Trees

Acrylic on canvas, 18" x 24", November, 2007.

I got into an en plein air thing again in the sunny winter of 2007. I wore a paint stained overcoat and wedged my paints into a knapsack and strapped a canvas to that. It was great to be all covered in paint clothes with a knapsack. I looked like a bum. People looked askance.

On a ridge in the Castle Rock park above Los Gatos and Saratoga I found some trees that were completely covered with fronds of moss. The sun was going down in the west over the Pacific Ocean, edging the mossy trees with brilliant yellow-green. The tree’s a little like a woman’s legs, too, very fertile. To pep up the picture, I added an eye. I like to wrap my paintings around, painting on the edges so I don’t have to frame them. I put another eye on the left edge, though you can’t see it in this image.

It was beautiful here. I was thinking of a drawing by Hieronymus Bosch were he sketches an eye on the ground and an ear on a tree.

32. Giant’s Head

Acrylic on canvas, 18" x 24", December, 2007.

Like “Mossy Trees,” “Giant’s Head” is a painting I started outdoors in Castle Rock Park near Saratoga, California. This particular rock is called California Ridge. I circled around on a narrow ledge to get to this vantage point. I was somewhat worried about falling off, there was a hundred foot drop to the ground. I painted my hand in there, like clutching at the rock to show that I was scared.

The rock itself reminded me of the profile of Homer Simpson. There were a lot of little lichen patches on it, I just tried to suggest those with some spots of color. That white line in the sky is a jet contrail. The green of the trees was really lovely, it felt good being alone up here on the ledge. I didn’t have room to stay very clean and I got a lot of paint on myself.

33. The Muse

Acrylic on canvas, 24" x 18", January, 2008.

My wife was out of town for a week visiting our daughter in New York, and I took my knapsack of paints and a canvas out to a cliff overlooking Four Mile Beach north of Santa Cruz, California. This spire of rock was probably part of a natural bridge many years ago. I often walk along the beach to this spot, it’s usually deserted and very beautiful. You don’t see any sign of human activity in any direction.

This was the first time I’d gotten onto the cliff right above the rock. It was a very windy day, and I found a depression in the cliff, a little grassy dell, and I settle in there. I particularly wanted to get the shape of the long, breaking dark wave near the horizon. A pelican flew past and I got a digital photo of him. I wished my wife were there with me.

When I got home, I kind of had to laugh at the inadequacy of the few daubs of paint I’d made—compared to the joyful, living seascape that I’d been looking at. It’s insane. You’re daubing ground up bits of stone onto a cloth and hoping to capture the physical world. But I did two more layers on the painting and finally I was happy with it.

To liven it up, I printed out a large image of the pelican I’d scene, also an image of my wife, and I slid those images around on the canvas until the composition looked right. And then I outlined those spots and painted copies of the images. I wasn’t sure I could do a human face—and the woman doesn’t really look that much like my wife. I think of her as “The Muse.” When I go out alone in nature, that’s who I’m hoping to hear from: the muse.

34. Spacetime Donuts

Acrylic on paper, 17" x 13", April, 2008.

In April of 2008, I arranged for a small press to reprint two of my early science fiction novels, Spacetime Donuts and The Sex Sphere. As part of the deal, they agreed to let me design the covers.

Spacetime Donuts is about a somewhat punk-like young mathematician who finds a way to shrink down so small that he wraps around the scale axis and gets big. Scale turns out to be circular, and spacetime is in some sense like a donut. I wrote this novel in 1979, and it can be argued that this was one of the very first cyberpunk science-fiction novels. The characters in the book plug their brains into computers, which is why I have that wire coming out of his neck.

He’s wearing an earring that’s a variation on the W.A.S.T.E. symbol in Thomas Pynchon’s novel The Crying of Lot 49. As it turns out, the book’s plot resembles this symbol. I had fun making this image really pop with cadmium red and cadmium yellow.

35. The Sex Sphere

Acrylic on canvas, 14" x 18", April, 20087.

In April of 2008, I arranged for a small press to reprint two of my early science fiction novels, Spacetime Donuts and The Sex Sphere. As part of the deal, they agreed to let me design the covers.

The Sex Sphere is about a being from the fourth dimension named Babs. Her intersection with our 3D space looks like parts of a woman, squeezed together and rounded off. She manipulates some of the characters into setting off a terrorist A-bomb in Florence, Italy. You can see the mushroom cloud in the background. I liked painting this, as it’s so intense and cartoony and surreal. I think the sex sphere looks a little scary.

Originally this painting was going to be a landscape looking out over Silicon Valley. I went up on St. Joseph’s Hill with a canvas and paints and started the picture there with my painter friend Vernon Head. Vernon knows my working habits by now, and he knew something weird was going to show up in the foreground. For awhile I wasn’t sure what it should be, but when I realized I needed a cover image for The Sex Sphere, I was ready to go.

Click here to see an earlier blog description of my first thirty paintings.

This entry follows on my two previous ones:
“Is the Universe Infinite?”
“Dialog on ”˜Is the Universe Infinite?’”

I just finished reading a great book, The Endless Universe . It’s by these two highly respected physics guys Paul Steinhardt and Neil Turok .

The Endless Universe argues that the Big Bang / Inflationary cosmological scenario has too many kludges to be plausible anymore. Till reading the book, I hadn’t realized how arbitrary and patched-up the inflationary scenario is by now. Cosmology is really a mess.

Steinhardt and Turok propose a simpler scenario called the cyclic universe or the “ekpyrotic” (Greek for “make of fire”) scenario under which we have two parallel branes (3d hyperplanes, that is, spaces like the universe), and every trillion years the branes spring together and FLASH all of space is filled with energy, and then the branes move apart, but not very far, only about a Planck length apart. Then they stop moving apart and hang there for a trillion years.

While hanging there, the branes expand, producing the galactic recession. The branes are like infinite planes, so they can expand exponentially and always have more room. And then, after a trillion years, they spring back together. The space in between them is sometimes known as “the bulk.” Strictly speaking the branes are 9-dimensional, with 3D for space and the 6 extra dimensions for the curled up and verminous Calabi-Yau-manifold stringy subdimensions. And the bulk dimension in between the branes is the 10th dimension.

There have been, in principle, an infinite number of cycles, that way we don’t need to face explaining the FIRST one. Each tortoise stands on the back of a previous tortoise.

I’d thought the cyclic universe theory had been discredited, but it’s still going strong, and fighting for more air. The old guard of inflation is fighting hard against the cyclic model, not even taking it very seriously. The Endless Universe is somewhat tendentious, arguing hard for the cyclic model over and over. But it convinced me.

Looking on the web, I find there’s a newer Baum-Frampton cyclic model mentioned on Wikipedia . In this model they talk about a cosmic “Big Rip,” which was formulated by some other guys in a paper called “Phantom Energy and Cosmic Doomsday.” The Big Rip is a depressing notion: the idea is that maybe space’s expansion is accelerating so wildly that eventually the expansion overcomes gravity and the nuclear forces so that first Earth is torn to bits, and then our bodies and even our atoms and the elementary particles. No, no, I much prefer the Cyclic Univere. It’s kind of wild these days, how you can find all these far-out science papers online.

One touchy bit in the Steinhardt-Turok model is the odd moment when the branes collide. To my amateur eye, it seems like it might be nicer and more symmetric if the branes passed through each other instead of bouncing. By the way, the branes are, strictly speaking orbifolds, whatever that means…the concept uses some insane juicy buzzwords like “quiver diagrams”.

Some bits from The Endless Universe that impacted my SF sensors:

(1) In seeming violation of the law of conservation of energy, a universe is allowed to “borrow” more energy from the gravitational field with each cycle. Every trillion years, some gravity gets converted into yet more mass and energy—the former mass and energy having been squandered by expansion of the brane space.

(2) Gravity waves are the one thing that might survive from one cycle to the next. But, methinks there may be other vestiges of the previous cycles that are kicked up (down?) into subtler planes; it helps if, as I like to do, we suppose matter to be infinitely divisible. Suppose someone can hear these “voices in the white.”

(3) They mention that if you have a collapse of space that’s not controlled carefully, you get wild asymmetric oscillations where, like, you turn into a million-mile-long cigar or a sheet of paint the size of the solar system: the chaotic mixmaster scenario.

(4) They really dump on the anthropic principle, which one might present as something like this: “A huge number of possible universes exists, and there just happens to be one that has its constants tuned in just the right way to support fleshapoids on a planet orbiting a sun.” I hadn’t realize how desperate and bankrupt the Big Bang/Inflation model had become. Supposedly there are scads of inflating universe and we just happen to be in this pariticular just-right one. It’s much more intellectually satisfying to suppose there’s only one universe and that there are some deep reasons for its properties.

[Note the small profile on the right, my ripple in the Big Flash.]

(5) In either model (inflation or cyclic), it’s been about 14 billion years since the, uh, call it the “Big Flash,” when space was filled with something like white light (higher energy than light, actually). The image of that flash gets in the way of seeing more than 14 billion light years away. But in the cyclic model, the space is in fact endless, and there’s lots of galaxies out past the 14 billion light year haze. I’m supposing (SFictionally speaking) that we can see past the haze via (a) faint gravity waves or (b) subtle energies relating to the as yet unknown subtler levels of matter. Supposedly we have another trillion years to go before the collapse, before the Big Splat that produces the next Big Flash.

(6) During the initial phase right after the Big Flash, a ripple that’s only a few meters across can in fact serve as the seed for a galaxy. Hmm…

[Adult ripple becomes Joey Ramone / T. Rex]

I might write a story about all this: “Voices in the White.”

Today I’m presenting an email dialog I had with Jeff Weeks about the whether the space of our universe is infinite.

Rudy: I mentioned in my most recent blog post, “Is the Universe Infinite” , that there seem to be three notions of space’s size. (i) it’s finite (ii) it’s potentially infinite, in that it’s finite but will expand forever (iii) it’s actually infinite. As a one-time set theorist, I of course would prefer (iii). But I don’t read much about this option.

Jeff: The reason you don’t hear much about that is that cosmologists don’t like to talk about things that can’t be tested against observation. So, for example, they are willing to consider the hypothesis that the universe is smaller than our horizon sphere (in which case we could in principle see repeating images and confirm the finiteness of space). But if and when it’s established that the universe is sufficiently larger than our horizon sphere, then many (but not all) cosmologists lose interest in discussing whether its truly infinite or just very, very big.

Rudy: In a quick web search, I see some (non-academic) people arguing against infinite universes on more a priori reasons, but these seem simply to be rehashes of pre-Cantorian Scholastic arguments derived from the mistaken notion that an actual infinite is inherently contradictory.

Jeff: Most (but not all) cosmologists avoid stating an opinion one way or the other (re a finite or infinite universe). They take the view that if you can’t test a hypothesis against observations, then it’s not worth discussing. In my view that approach seems a little extreme, but on the other hand I can see where they are coming from. Many physicists feel burnt (or at least chastened) by quantum mechanics, leading them to seek refuge in the idea that the purpose of science is to predict the results of experiments, and that scientists shouldn’t waste their time speculating about what it all means. I don’t agree with that approach, but like I said I can see where they’re coming from. Anyhow, for that reason I think most cosmologists are agnostic on the question of a finite or infinite universe.

Rudy: I guess you’d need to be in 4D space to smoothly make the 3D torus or the Poincare dodecahedron.

Jeff: We humans like things to sit in Euclidean spaces, because we can imagine them more easily that way. But Mother Nature suffers no such limitations. Thus it makes perfect sense to imagine a 3-torus that doesn’t sit in 4D space (and in fact doesn’t sit in *any* space). It just is. It’s itself, and that’s that.

By the way, one of my motivations for writing the Torus and Klein Bottle Games was to let users develop some gut-level intuition for a finite multi-connected space that doesn’t sit in any higher-dimensional space. That is, when you play the games, you learn to understand the finite 2D surfaces “as themselves”, never having to wrap them around in 3- or 4-dimensional space.

Rudy: I’m not fully clear on how to express the difference between the hypersphere and the positively curved finite compact Poincaré dodecahedral space. I’m guessing it’s that the Poincare space is analogous to a multiple-genus surface with holes in it? Six holes?

Jeff: Yes, exactly, the Poincare dodecahedral space is analogous to a multiple-genus surface, in the sense that both are “multiply connected”.

The possible shapes for a 3D space are far richer than the possibilities for a 2D surface. So while a simple concept like the number of holes (or, equivalently, the number of handles) works great for classifying 2D surfaces, it breaks down in 3D. In other words, the set of (orientable) 2D surfaces is linear in the sense that you can line ”˜em up in a row and not miss anything, like:

sphere, doughnut surface, 2-holed doughnut surface, etc.

But the set of possible 3D shapes isn’t linear in that way. A more productive way to think about 3D spaces is to focus on which directions you could travel to return to your starting point, or, equivalently, in which directions you could look and see an image of yourself. For example, in a 3-torus you’d see your nearest self-images along a set of three mutually perpendicular axes (i.e. if you’re standing at the center of a cubical fundamental domain, and you direct your gaze towards the center of any of the cube’s six faces, you’ll see an image of yourself “one unit away”). In the dodecahedral space, by contrast, you can look towards any of the dodecahedron’s 12 faces to see a nearest self-image. You can enjoy these effects first hand in my Curved Spaces software: just open a space of interest and use the left- and right-arrow keys to give your a little window to see across each of the walls.

Rudy: I’m also groping for a good way to describe some in-space scenario that would make the “holes” evident.

Jeff: Maybe something related to where you see your self-images? Another interesting twist in the plot would be that when you see those images, you’re seeing into the past (because of the finite speed of light).

One last comment here: In the hypersphere you see exactly one image of yourself, and in fills the whole sky. I think that observation was in your first book, Geometry, Relativity and the Fourth Dimension ? That book had a big influence on my intellectual development. It was a real hit among my math/science friends!

Rudy: Your paper “The Poincaré Dodecahedral Space and the Mystery of the Missing Fluctuations” suggests that, since we don’t see much in the way of a low mode gradient in the cosmic microwave background radiation (CMB) across the sky, our space is probably finite. But there are of course a number of ways out of this argument, right?

Jeff: Yes. The weak broad-scale CMB fluctuations could be a result of a multiconnected topology, or they could be the result of some other effect, or they could be just a statistical fluke.

Rudy: Funny, it seems so IMPORTANT to me whether there are alef-null stars, and to many this is a meaningless question. I gather that at least it’s not viewed as impossible, which is reassuring.

Jeff: It’s a natural human drive to want to understand the world we live in. What aspects of the world we find interesting are shaped by our previous experience. This is a healthy thing, I think, because it means that different people end up obsessed by different things (you with infinite sets, me with topology/geometry, and so on) and thus we avoid having everybody thinking alike. Thank goodness!

Rudy: I have a dream that eventually the transfinite will emerge into testable quotidien physics

Jeff: That’s the beautiful part — lots of people with lots of dreams.